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Wave propagation in anelastic rocks is a relevant scientific topic in basic research with applications in exploration geophysics. The classical Biot theory laid the foundation for wave propagation in porous media composed of a solid frame and a saturating fluid, whose constitutive relations are linear. However, reservoir rocks may have a high-viscosity fluid, which exhibits a non-Newtonian (nN) behaviour. We develop a poroelasticity theory, where the fluid stress-strain relation is described with a Kelvin–Voigt mechanical model, thus incorporating viscoelasticity. First, we obtain the differential equations from first principles by defining the strain and kinetic energies and the dissipation function. We perform a plane-wave analysis to obtain the wave velocity and attenuation. The validity of the theory is demonstrated with three examples, namely, considering a porous solid saturated with a nN pore fluid, a nN fluid containing solid inclusions and a pure nN fluid. The analysis shows that the fluid may cause a negative velocity dispersion of the fast P (S)-wave velocities, that is velocity decreases with frequency. In acoustics, velocity increases with frequency (anomalous dispersion in optics). Furthermore, the fluid viscoelasticity has not a relevant effect on the wave responses observed in conventional field and laboratory tests. A comparison with previous theories supports the validity of the theory, which is useful to analyse wave propagation in a porous medium saturated with a fluid of high viscosity. [ABSTRACT FROM AUTHOR]

In principle, wave propagation in porous media can simultaneously trigger macroscopic fluid flow and thermal flow, which can be described by Biot's poroelasticity and Lord–Shulman thermoelasticity, respectively. The physical processes of those effects are significantly different, but phenomenologically, they can lead to identical wave attenuation and dispersion and are hard to be distinguished. By using Biot's virtual concept entropy flow, the Biot-consistent General Linear Solid (the GLS) framework and matrix notation, a rigorous and convenient tool is provided to reveal the similarities and disparities between poroelasticity and thermoelasticity. By using the same framework, a Biot-consistent thermo-poroelastic model is proposed to consider macroscopic effects of fluid and thermal flows simultaneously in an elegant way. These similarities allow us to directly translate many of the available results in poroelasticity to thermoelasticity and vice versa by a simple change of notation. The disparities indicate a fundamental difference in physical mechanisms. Plane-wave analysis shows that the primary P -wave modes of thermoelasticity and poroelasticity are all GSLS-equivalent (Generalized Standard Linear Solid) and can be identical if the model parameters are selected properly. However, the corresponding slow-wave modes have significantly different phase velocity dispersion although the attenuation spectra of which are identical. Such a surprising result can be explained by the GSLS non-equivalence of the slow-wave modes and the fundamentally different mechanisms. As expected, the thermo-poroelastic model predicts four wave modes, which are the fast- and slow- P , temperature (T wave) and S waves. Two attenuation peaks due to, respectively, the thermal- and fluid-flow effects are predicted for the fast- P wave. The slow- P wave mode due to fluid flow is influenced by the thermal effects, but the T wave seems unaffected by the fluid flow. The thermo-poroelastic model is then applied to laboratory observations at 200–106 Hz for the brine-saturated tight sandstone under 35 MPa effective pressure. The unified model provides a convenient framework for studying geothermal exploration, thermal-enhanced oil recovery and other applications involving temperature variations within the porous rock. [ABSTRACT FROM AUTHOR]

The theory of wave propagation in non-isothermal porous rocks has been introduced in geophysics in recent years by combining the single-phase-lag (SPL) Lord-Shulman (LS) model of heat conduction with Biot's poroelasticity theory. However, the theory of SPL thermoporoelasticity is inadequate in describing the lagging behavior of thermal relaxation for wave dissipation due to fluid and heat flow effects. We address this problem by incorporating a dual-phase-lags (DPL) model of heat conduction into thermoporoelasticity, utilizing analytical solutions and numerical simulations. The DPL model involves two lagging times: the (macroscopic) heat-flux lagging time τq from the LS model and an additionally introduced lagging time τT of temperature gradient that characterizes the fluid phase. A plane-wave analysis predicts four propagation waves, namely, fast P, slow P, thermal (T), and shear (S). We calculate wavefield snapshots by using a finite-difference solver for the DPL thermoelastic equations and provide further insight into the physics of two lagging times for porous rocks. The simulations show that the DPL model induces higher thermal attenuation and larger velocity dispersion compared to the SPL model, especially at high frequencies. The influence of fluids is crucial for wave propagation within thermoporoelastic media. [ABSTRACT FROM AUTHOR]

Seismic exploration of deep oil/gas reservoirs involves the propagation of seismic waves in high-pressure media. Traditional elastic wave equations are not suitable for describing such media. The theory of acoustoelasticity establishes the dynamic equation of wave propagating in prestressed media through constitutive relation using third-order elastic constants. Many studies have been carried out on numerical simulations for acoustoelastic waves, but are mainly limited to 2D cases. A standard staggered-grid (SSG) finite-difference (FD) approach and the perfectly matched layer (PML) absorbing boundary are combined to solve 3D first-order velocity-stress equations of acoustoelasticity to simulate wave propagating in 3D prestressed solid medium. Our numerical results are partially validated by plane-wave analytical solution through the comparison of calculated and theoretical P-/S-wave velocities as a function of confining prestress. We perform numerical simulations of acoustoelastic waves under confining, uniaxial and pure-shear prestressed conditions. The results show the stress-induced velocity anisotropy in acoustoelastic media, which is closely related to the direction of prestresses. Comparisons to seismic simulations based on the theory of elasticity illustrate the limitation of conventional elastic simulations for prestressed media. Numerical simulations prove the significant effect of prestressed conditions on seismic responses. [ABSTRACT FROM AUTHOR]

Attenuation is often significant during seismic wave propagation in the subsurface, leading to the reduced resolution and narrower bandwidth of seismic images. Traditional corrections for such effects are inverse-Q filtering and deconvolution, which require a high signal-to-noise ratio (SNR) to avoid noise boost-up. Here, we propose a time-domain method offering advantages in the resolution and interpretational quality of the resulting images. Similar to wavelet transforms, the iterative time-domain deconvolution (ITD) represents the seismogram by a superposition of non-stationary source wavelets modeled in the appropriate attenuation model. Arbitrary frequency-dependent Q and velocity dispersion laws can be used and non-Q type attenuation can be caused by focusing, defocusing, scattering, effects of fine layering, and fluctuations of the wavefield. Compared to inverse-Q filtering and some deconvolution methods, the method does not boost high-frequency noise and is less sensitive to the accuracy of the Q model. We illustrate and compare this method to inverse-Q filtering by using several synthetic and real data examples. The tests include noise-contaminated data, inaccurate Q models, and variable source wavelets. The examples show that the ITD is a practical and effective tool for Q-compensation with a broad scope of potential applications, albeit with some defects. An important benefit of ITD that other methods may not possess could be the ability to utilize geological information, such as locations and sparseness of major reflectors or the presence of interpreted Q contrasts, which might be able to further improve the performance of ITD. [ABSTRACT FROM AUTHOR]

Thermoelastic attenuation of P waves is due to energy conversion to the heat mode, which is diffusive at low frequencies and wave-like at high frequencies, behaving similarly to the Biot slow mode. The conversion is strong in highly heterogeneous media. We consider a layered medium with a random distribution of thermal properties, specifically the Grüneisen ratio, and obtain the phase velocity and quality factor. The relaxation peak of the random medium is wider than those of a periodic sequence of layers and the Zener mechanical model. Indeed, a Cole–Cole fractional model is needed to obtain a good match. These approximations are required to compute wave fields in heterogeneous media. Moreover, the solutions are helpful for studying the physics of thermoelasticity and testing numerical algorithms for wave propagation. [ABSTRACT FROM AUTHOR]

Poro-acoustoelastic theory has made a great progress in both theoretical and experimental aspects, but with no publications on the joint research from theoretical analyses, experimental measurements, and numerical validations. Several key issues challenge the joint research with comparisons of experimental and numerical results, such as digital imaging of heterogeneous poroelastic properties, estimation of acoustoelastic constants, numerical dispersion at high frequencies and strong heterogeneities, elastic nonlinearity due to compliant pores, and contamination by boundary reflections. Conventional poro-acoustoelastic theory, valid for the linear elastic deformation of rock grains and stiff pores, is modified by incorporating a dual-porosity model to account for elastic nonlinearity due to compliant pores subject to high-magnitude loading stresses. A modified finite-element method is employed to simulate the subtle effect of microstructures on wave propagation in prestressed digital cores. We measure the heterogeneity of samples by extracting the autocorrelation length of digital cores for a rough estimation of scattering intensity. We conductexperimental measurements with a fluid-saturated sandstone sample under a constant confining pressure of 65 MPa and increasing pore pressures from 5 to 60 MPa. Numerical simulations for ultrasound propagation in the prestressed fluid-saturated digital core of the sample are followed based on the proposed poro-acoustoelastic model with compliant pores. The results demonstrate a general agreement between experimental and numerical waveforms for different stresses, validating the performance of the presented modeling scheme. The excellent agreement between experimental and numerical coda quality factors demonstrates the applicability for the numerical investigation of the stress-associated scattering attenuation in prestressed porous rocks. [ABSTRACT FROM AUTHOR]

Thermoelastic attenuation is similar to wave-induced fluid-flow attenuation (mesoscopic loss) due to conversion of the fast P wave to the slow (Biot) P mode. In the thermoelastic case, the P - and S -wave energies are lost because of thermal diffusion. The thermal mode is diffusive at low frequencies and wave-like at high frequencies, in the same manner as the Biot slow mode. Therefore, at low frequencies, that is, neglecting the inertial terms, a mathematical analogy can be established between the diffusion equations in poroelasticity and thermoelasticity. We study thermoelastic dissipation for spherical and cylindrical cavities (or pores) in 2-D and 3-D, respectively, and a finely layered system, where, in the latter case, only the Grüneisen ratio is allowed to vary. The results show typical quality-factor relaxation curves similar to Zener peaks. There is no dissipation when the radius of the pores tends to zero and the layers have the same properties. Although idealized, these canonical solutions are useful to study the physics of thermoelasticity and test numerical algorithm codes that simulate thermoelastic dissipation. [ABSTRACT FROM AUTHOR]

A modified LSM is proposed by introducing an independent micro-rotational inertia, which may help characterize the scale-dependent effect and avoid the Poisson's ratio limitation of regular triangle lattices in two dimensions. For this method, some factors may affect data pickup and modeling accuracy, but the 'optimal' inputs, like stiffness ratio, numerical damping, and micro-rotational inertia, could be obtained from parameter identification by the Dakota toolkit (Adams et al., Tech Rep SAND2010–2183, 2009), when a suitable excitation source function and lattice spacing are set up. By comparing with the modified couple stress theory, we analyze the dispersion relationship of elastic waves for the estimation of the characteristic material length. It shows that this modified LSM may provide an alternative and promising way to investigate the size-dependent wave propagation in elastic media numerically. [ABSTRACT FROM AUTHOR]

The classical Zener model of thermoelasticity can be represented by a mechanical (or viscoelastic) model based on two springs and a dashpot, commonly called standard-linear solid, whose parameters depend on the thermal properties and a relaxation time, and yield the isothermal and adiabatic velocities at the low- and high-frequency limits. This model differs from the more general Lord-Shulman theory of thermoelasticity, whose low-frequency velocity is the adiabatic one. These theories are the basis of thermoelastic attenuation in inhomogeneous media, with heterogeneities much smaller than the wavelength, such as Savage's theory of thermoelastic dissipation in a medium with spherical pores. In this case, the shape of the relaxation peak differs from that of the Zener and Lord-Shulman models. In these effective homogeneous media, the anelastic behavior of real materials can better be described by using a stress-strain relation based on fractional derivatives. In particular, wave propagation (dispersion and attenuation) is well described by a Cole-Cole stress-strain equation, as illustrated by the agreement with Savage's theory. We propose a time-domain algorithm based on the Grünwald-Letnikov numerical approximation of the fractional time derivative involved in the time-domain representation of the Cole-Cole model. The spatial derivatives are computed with the Fourier pseudospectral method. We verify the results by comparison with the analytical solution, based on the Green function. The numerical example illustrates wave propagation at an interface separating a porous medium and a purely solid phase. [ABSTRACT FROM AUTHOR]

Thermoelasticity extends the classical elastic theory by coupling the fields of particle displacement and temperature. The classical theory of thermoelasticity, based on a parabolic-type heat-conduction equation, is characteristic of an unphysical behaviour of thermoelastic waves with discontinuities and infinite velocities as a function of frequency. A better physical system of equations incorporates a relaxation term into the heat equation; the equations predict three propagation modes, namely, a fast P wave (E wave), a slow thermal P wave (T wave), and a shear wave (S wave). We formulate a second-order tensor Green's function based on the Fourier transform of the thermodynamic equations. It is the displacement–temperature solution to a point (elastic or heat) source. The snapshots, obtained with the derived second-order tensor Green's function, show that the elastic and thermal P modes are dispersive and lossy, which is confirmed by a plane-wave analysis. These modes have similar characteristics of the fast and slow P waves of poroelasticity. Particularly, the thermal mode is diffusive at low thermal conductivities and becomes wave-like for high thermal conductivities. [ABSTRACT FROM AUTHOR]

We develop a numerical algorithm for simulation of wave propagation in linear nonisothermal poroelastic media, based on Biot theory and a generalized Fourier law of heat transport in analogy with Maxwell model of viscoelasticity. A plane wave analysis indicates the presence of the classical P and S waves and two slow waves, namely, the Biot and the thermal slow modes of propagation, which present diffusive behavior under certain conditions, depending on viscosity, frequency, and the thermoelastic constants. The wavefield is computed with a direct meshing method using the Fourier differential operator to calculate the spatial derivatives. We propose two alternative time‐stepping algorithms, namely, a first‐order explicit Crank‐Nicolson method and a second‐order splitting method. The Fourier differential operator provides spectral accuracy in the calculation of the spatial derivatives. Modeling the thermal diffusive mode is relevant for high‐temperature high‐pressure fields and since it leads to mesoscopic attenuation by mode conversion of the fast waves to the thermal waves. Key Points: The differential equations of dynamic thermoporoelasticity are solved with a direct grid methodThe solver is based on the Fourier pseudospectral method to compute the spatial derivativesThe simulation reveals the behavior of four wave modes, namely, the fast P, slow P, S, and the thermal wave-diffusion field [ABSTRACT FROM AUTHOR]

Abstract: For wave propagation simulation in piecewise heterogeneous media, Gaussian-elimination-based full-waveform solutions to the generalized Lippmann–Schwinger integral equation (GLSIE) are highly accurate, but involved with extremely time-consuming computations because of the very large size of the resulting boundary–volume integral equation matrix to be inverted. Several flexible approximations to the GLSIE are scaled in an iterative way to adapt numerical solutions to the smoothness of heterogeneous media in terms of incident wavelengths, with a great saving of computing time and memory. Among various typical iterative schemes to the GLSIE matrix, the generalized minimal residual method (GMRES) is an efficient approach to reduce the computational intensity to some degree. The most efficient approximation can be obtained using a Born series, as an alternative iterative solution, to both the boundary-scattering and volume-scattering waves, leading to the Born-series approximation (BSA) scheme and the improved Born-series approximation (IBSA) scheme. These iteration schemes are validated by dimensionless frequency responses to a heterogeneous semicircular alluvial valley, and then applied to a heterogeneous multilayered model by calculating synthetic seismograms to evaluate approximation accuracies. Numerical experiments, compared with the full-waveform numerical solution, indicate that the convergence rates of these methods decrease gradually with increasing velocity perturbations. The comparison also shows that the BSA scheme has a faster convergence than the GMRES method for velocity perturbations less than 10 percent, but converges slowly and even hardly achieves convergence for velocity perturbations greater than 15 percent. The IBSA scheme gives a superior performance over the other methods, with the least iterations to achieve the necessary convergence. [Copyright &y& Elsevier]

The computation of derivatives in governing partial differential equations is one of the most investigated subjects in the numerical simulation of physical wave propagation. An analytical staggered-grid convolutional differentiator (CD) for first-order velocity-stress elastic wave equations is derived in this paper by inverse Fourier transformation of the band-limited spectrum of a first derivative operator. A taper window function is used to truncate the infinite staggered-grid CD stencil. The truncated CD operator is almost as accurate as the analytical solution, and as efficient as the finite-difference (FD) method. The selection of window functions will influence the accuracy of the CD operator in wave simulation. We search for the optimal Gaussian windows for different order CDs by minimizing the spectral error of the derivative and comparing the windows with the normal Hanning window function for tapering the CD operators. It is found that the optimal Gaussian window appears to be similar to the Hanning window function for tapering the same CD operator. We investigate the accuracy of the windowed CD operator and the staggered-grid FD method with different orders. Compared to the conventional staggered-grid FD method, a short staggered-grid CD operator achieves an accuracy equivalent to that of a long FD operator, with lower computational costs. For example, an 8th order staggered-grid CD operator can achieve the same accuracy of a 16th order staggered-grid FD algorithm but with half of the computational resources and time required. Numerical examples from a homogeneous model and a crustal waveguide model are used to illustrate the superiority of the CD operators over the conventional staggered-grid FD operators for the simulation of wave propagations. [ABSTRACT FROM AUTHOR]

Wave propagation in rocks induces fluid pressure gradients and flow between stiff intergranular pores and compliant microfractures. This, in turn, leads to wave anelasticity, i.e., attenuation and velocity dispersion. Three known squirt-flow models attempt to explain this phenomenon. Based on these theories, we reformulate a new one, where we derive the dry-rock moduli on the assumption that the boundary fluid pressure is constant. The new model improves the predictions of the other models, e.g., a better description at high frequencies and the inclusion of permeability effects. The numerical example shows the response due to the characteristic squirt-flow length and fluid viscosity on wave propagation. The reformulated model is applied to experiments made on water-saturated tight sandstones, where the squirt-flow length is determined, showing that it increases with increasing permeability and porosity at full saturation. • A new squirt-flow model by deriving the dry-rock moduli on the assumption of the constant boundary fluid pressure. [ABSTRACT FROM AUTHOR]

The double-porosity theory of wave propagation in rocks considers a single scale based on soft and stiff pores, and wave attenuation occurs by local fluid flow between these pores. However, rocks exhibit more complicated fabric structures, with heterogeneities presented at multiple scales, having a self-similar fractal nature as reported. We develop a poroelasticity model to describe wave anelasticity in a fluid-saturated medium consisting of infinite components, i.e., an infinituple-porosity medium. Numerical modeling is achieved with infinite iterations, where at each iteration one component is added (a porous inclusion embedded in a porous host), analogous to the differential effective medium (DEM) theory of solid composites. By considering the self-similar structure, the properties of each component (inclusions) are scale-dependent. The example shows that the P-wave velocity dependence on frequency depends on the fractal dimension. Application of the model to sandy sea-bottom sediments shows a good agreement, with fractal dimension of 2.88. [ABSTRACT FROM AUTHOR]